Suppose root 5 is a rational numberTherefore root 5= p/ , p,q are integers, & q is not equals to 0On reducing– root 5= a/b , a & b are coprime no.=> b root 5= aSquaring both sides5b^2= a^2 ————(i)=> b^2 = a^2 /5Since 5 divides a^2Therefore 5 will also divide a (by theorem 1.3) —-(ii)=> a/5= c=> a= 5c=> a^2 = 25c^2=> 5b^2 = 25c^2 (by eq.^n (i))=> b^2= 5c^2=> b^2 /5= c^2Therefore 5 divides b^2 Therefore 5 will also divide b (by theorem 1.3) —–(iii)From eq.^n (ii) & (iii)5 divides a & b bothTherefore 5 is common factor of a & bBut this contradict the fact that a & b are coprimeTherefore this is due to our wrong assumption that root 5 is rational.Therefore we conclude that root 5 is rational.